= Solution
The <spectral theorem> for a <Hermitian matrix> gives a unitary diagonalizer $V$ with $V\Phi V^\dagger=\operatorname{diag}(x_1,\ldots,x_5)$ and real $x_i$. Similarity preserves the <trace>, so $\sum_i x_i=0$.
To ensure that the diagonalizer belongs to <SU(5)> rather than only the <unitary group> $U(5)$, write $\det V=e^{i\alpha}$ and put $U=e^{-i\alpha/5}V$. Its determinant is one, and the scalar phase cancels from its conjugation action. Thus
$$
\boxed{U\Phi U^\dagger=\Phi_d=\operatorname{diag}(x_1,\ldots,x_5),\qquad
x_i\in\mathbb R,\quad\sum_i x_i=0,\quad U\in SU(5).}
$$
The same argument shows why no additional reality condition beyond Hermiticity is needed. Permuting the real <eigenvalues> merely changes their order in the diagonal form.
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